Ab initio molecular dynamics: Theory and Implementation

¢¢¢¢¢i = 1 . . .N b¢c i (G)¢ inv FTc i (R)V loc (R)c i (G)c i (R)inv FTn(R) ← |c i (R)| 2n(R)FC i (R) = V loc (R)c i (R)FC i (G)fw FTFigure 8. Flow chart for the calculation of the charge density (on the left) and the force on thewavefunction from the local potential (on the right). The charge density calculation requires N b(number of states) three dimensional Fourier transforms. For the application of the local potentialtwo Fourier transforms per state are needed. If enough memory is available the first transform canbe avoided if the wavefunction on the real space grid are stored during the density calculation.potential (see also Fig. 7) and the Poisson solver in cases when the Hockney methodis used (see Fig. 6).The calculation of the total energy, together with the local potential is shownin Fig. 10. The overlap between the projectors of the nonlocal pseudopotential andthe wavefunctions calculated in this part will be reused in the calculation of theforces on the wavefunctions. There are three initialization steps marked in Fig. 9.Step one has only to be performed at the beginning of the calculation, as thequantities g and E self are constants. The quantities calculated in step two dependon the absolute value of the reciprocal space vectors. They have to be recalculatedwhenever the box matrix h changes. Finally, the variables in step three dependon the atomic positions and have to be calculated after each change of the nuclearpositions. The flow charts of the calculation of the forces for the wavefunctions andthe nuclei are shown in Figs. 11 and 12.3.6 Optimizing the Kohn-Sham OrbitalsAdvances in the application of plane wave based electronic structure methods areclosely related to improved methods for the solution of the Kohn–Sham equations.There are now two different but equally successful approaches available. Fix–pointmethods based on the diagonalization of the Kohn–Sham matrix follow the moretraditionally ways that go back to the roots of basis set methods in quantum chemistry.Direct nonlinear optimization approaches subject to a constraint were initiatedby the success of the Car–Parrinello method. The following sections reviewsome of these methods, focusing on the special problems related to the plane wavebasis.63