Ab initio molecular dynamics: Theory and Implementation

and a matrix form of the Gram–Schmidt procedureX ji = (G T ) −1ji , (226)where S is the overlap matrix and G is its Cholesky decompositionS = GG T . (227)Recently new methods that avoid the orthogonalization step have been introduced.One of them 483 relies on modified functionals that can be optimized withoutthe orthogonality constraint. These functionals, originally introduced in the contextof linear scaling methods 417,452 , have the property that their minima coincidewith the original Kohn–Sham energy functional. The methods described above canbe used to optimize the new functional.Another approach 309 is to use a variable transformation from the expansioncoefficients of the orbitals in plane waves to a set of non–redundant orbital rotationangles. This method was introduced in quantum chemistry 618,149,167 and is usedsuccessfully in many optimization problems that involve a set of orthogonal orbitals.A generalization of the orbital rotation scheme allowed the application also for caseswhere the number of basis functions is orders of magnitudes bigger than the numberof occupied orbitals. However, no advantage is gained over the standard methods,as the calculation of the gradient in the transformed variables scales the same as theorthogonalization step. In addition, there is no simple and efficient preconditioneravailable for the orbital rotation coordinates.3.6.4 Fix-Point MethodsOriginally all methods to find solutions to the Kohn–Sham equations were usingmatrix diagonalization methods. It became quickly clear that direct schemes canonly be used for very small systems. The storage requirements of the Kohn–Shammatrix in the plane wave basis and the scaling proportional to the cube of the basisset size lead to unsurmountable problems. Iterative diagonalization schemes can beadapted to the special needs of a plane wave basis and when combined with a properpreconditioner lead to algorithms that are comparable to the direct methods, bothin memory requirements and over all scaling properties. Iterative diagonalizationschemes are abundant. Methods based on the Lanczos algorithm 357,151,489 canbe used as well as conjugate gradient techniques 616,97 . Very good results havebeen achieved by the combination of the DIIS method with the minimization ofthe norm of the residual vector 698,344 . The diagonalization methods have to becombined with an optimization method for the charge density. Methods based onmixing 153,4 , quasi-Newton algorithms 92,77,319 , and DIIS 495,344,345 are successfullyused. Also these methods use a preconditioning scheme. It was shown that the optimalpreconditioning for charge density mixing is connected to the charge dielectricmatrix 153,4,299,658,48 . For a plane wave basis, the charge dielectric matrix can beapproximated by expressions very close to the ones used for the preconditioning inthe direct optimization methods.Fix-point methods have a slightly larger prefactor than most of the direct methods.Their advantage lies in the robustness and capability of treating systems withno or small band gaps.69