Pulay stress

For more information on Pulay stress and related complications associated with finite sets of plane waves and k points when calculating forces in supercells with varying volumes, see G. P. Francis and M. C. Payne, /. Phys. Condens. Matter 2 (1990), 4395. 

The arguments of W are explicitly shown here to stress that this equation is valid only for the correct variational solution C(R). Eq. (5) shows the important fact that the first-order changes in the variational parameters are not needed for the evaluation of the gradients of a variational energy expression. This was already stressed by an early publication of Pulay (1969). The importance of this lies in the fact that the solution of the first-order response equations... 

All calculations have been performed taking fixed positions for the atoms in the structures, so that no relaxations are considered. The Siesta method, however, allows relaxations and molecular dynamics by calculating the forces on the atoms and the stress tensor from the Hellman-Feynman theorem with Pulay corrections The computational cost would be much higher than in the frozen geometry calculations, but the calculations for complex magnetic systems are cheaper than with other ab initio methods (KKR, FLAPW, LMTO). Apart from the use of pseudopotentials and a minimal basis set, the supercell construction does not require the inclusion of empty spheres as it is the case for the LMTO-ASA method. Besides, going from 2D to 3D supported clusters of similar size does not increase the computational cost like in the KKR-GF method where Green-Functions have to be computed. 

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