Ab initio molecular dynamics: Theory and Implementation

and the corresponding potential isThe sum of all modified core densitiesV xc = V xc (n + ñ core ) . (203)ñ core (G) = ∑ Iñ I core(G)S I (G) (204)depends on the nuclear positions, leading to a new contribution to the forcesand to the stress tensor∂E xc∂R I,s= −Ω ∑ G∂E xc∂h uv= ∑ I∑GiG s V ⋆xc(G)ñ I core(G)S I (G) , (205)V ⋆xc(G) ∂ñI core(G)∂h uvS I (G) . (206)The derivative of the core charge with respect to the cell matrix can be performed inanalogy to the formula given for the local potential. The method of the non-linearcore correction dramatically improves results on systems with alkali and transitionmetal atoms. For practical applications, one should keep in mind that the nonlinearcore correction should only be applied together with pseudopotentials thatwere generated using the same energy expression.3.5 Energy and Force Calculations in PracticeIn Sect. 3.4 formulas for the total energy and forces were given in their Fourierspace representation. Many terms are in fact calculated most easily in this form,but some terms would require double sums over plane waves. In particular, thecalculation of the charge density and the wavefunction gradient originating fromthe local potential∑Vloc(G ⋆ − G ′ )c i (G ′ ) . (207)G ′The expression in Eq. (207) is a convolution and can be calculated efficiently by aseries of Fourier transforms. The flow charts of this calculations are presented inFig. 8. Both of these modules contain a Fourier transform of the wavefunctions fromG space to the real space grid. In addition, the calculation of the wavefunctionforces requires a back transform of the product of the local potential with thewavefunctions, performed on the real space grid, to Fourier space. This leads toa number of Fourier transforms that is three times the number of states in thesystem. If enough memory is available on the computer the second transform ofthe wavefunctions to the grid can be avoided if the wavefunctions are stored in realspace during the computation of the density. These modules are further used inthe flow chart of the calculation of the local potential in Fig. 9. Additional Fouriertransforms are needed in this part of the calculation. However, the number oftransforms does not scale with the number of electrons in the system. Additionaltransforms might be hidden in the module to calculate the exchange and correlation62