Computation of Forces on the Ions

Equation (15) for the ground-state electron density can be differentiated with respect to the atomic coordinates to compute the forces acting on the atoms 

This term vanishes under either of the following conditions (1) the functional E[p] is stationary with respect to variations of the KS orbitals, i.e. the [ i] are indeed eigenstates of the KS Hamiltonian //ks (2) the derivative d r)/dRi is identically zero. 

Owing to the incompleteness of the basis and to numerical errors, the first condition is never met with a sufficient degree of accuracy in practical computations. The second condition is less demanding the d l/ (r)/dRi partial derivative has to vanish unless the basis functions depend explicitly on the atomic positions. This is the case, for instance, for localized basis sets centered on the atoms [79]. Plane-waves, on the other hand, do not depend on the atomic coordinates, and therefore the force contribution given in Eq. (24) vanishes. In this case the computation of the forces is particularly simple, as it is reduced to the second contribution of the right-hand side of Eq. (23)  

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