Ab initio molecular dynamics: Theory and Implementation

¢¢¢¢¢c i (G)123g, E self∆V Ilocal (G), nI c(G), P I α(G)S I (G), E ovrl∆V local (G), n c (G)local potentialmoduleF α I,iE kinE PPnlE tot , V loc (R)Figure 10. Flow chart for the calculation of the local potential and total energy. Initializationsteps are marked with numbers. Step 2 has to be repeated whenever the size of the unit cellchanges. Step 3 has to be repeated whenever nuclear positions have changed.in plane waves and using the same routines as in the standard code the Kohn–Shamand overlap matrices are calculated in this basis. The general eigenvalue problem issolved and the eigenfunctions can easily be expressed in the plane wave basis thatare in turn used as the initial wavefunctions to the optimization routines. Similarly,a given plane wave representation of the total wavefunction can be projected onto anauxiliary set of atom–centered functions. This opens up the possibility to performpopulation and bond–order analyses (following for instance the schemes of Mullikenor Mayer) in plane wave–pseudopotential calculations 537 .3.6.2 PreconditioningOptimizations in many dimensions are often hampered by the appearance of differentlength scales. The introduction of a metric that brings all degrees of freedomonto the same length scale can improve convergence considerably. The applicationof such a metric is called ”preconditioning” and is used in many optimizationproblems. If the variables in the optimization are decoupled the preconditioningmatrix is diagonal and becomes computationally tractable even for very large systems.Fortunately, this is to a large degree the case for a plane wave basis set. Forlarge G vectors the Kohn–Sham matrix is dominated by the kinetic energy whichis diagonal in the plane wave representation. Based on this observation efficient65