where k ∈ {1, ..., N P }. G s (Ω * s ) and E s (U * inter,s ) are algebraic functions of Ω * s and U * inter,s , respectively, but Ω * s and U * inter,s are obtained from the solution of a local minimisation and do not have a closed form expression. Nevertheless, analytical expressions for the derivatives of these variables with respect to pk can be obtained by manipulation of the optimality conditions of the lattice minimisation defined in Problem (LM). The full details the derivations can be found elsewhere . An outline of the derivations along with the final equations are supplied here.