Ab initio molecular dynamics: Theory and Implementation

disposable parameters that can be optimized for a particular calculation. Note thatthe discretization points in continuous space can also be considered to constitute asort of “finite basis set” – despite different statements in the literature – and thatthe “infinite basis set limit” is reached as h → 0 for N fixed. A variation on thetheme are Mehrstellen schemes where the discretization of the entire differentialequation and not only of the derivative operator is optimized 89 .The first real–space approach devised for ab initio molecular dynamics was basedon the lowest–order finite–difference approximation in conjunction with a equally–spaced cubic mesh in real space 109 . A variety of other implementations of moresophisticated real–space methods followed and include e.g. non–uniform meshes,multigrid acceleration, different discretization techniques, and finite–element methods686,61,39,130,131,632,633,431,634 . Among the chief advantages of the real–spacemethods is that linear scaling approaches 453,243 can be implemented in a naturalway and that the multiple–length scale problem can be coped with by adapting thegrid. However, the extension to such non–uniform meshes induces the (in)famousPulay forces (see Sect. 2.5) if the mesh moves as the nuclei move.3 Basic Techniques: Implementation within the CPMD Code3.1 Introduction and Basic DefinitionsThis section discusses the implementation of the plane wave–pseudopotential moleculardynamics method within the CPMD computer code 142 . It concentrates on thebasics leaving advanced methods to later chapters. In addition all formulas are forthe non-spin polarized case. This allows to show the essential features of a planewave code as well as the reasons for its high performance in detail. The implementationof other versions of the presented algorithms and of the more advancedtechniques in Sect. 4 is in most cases very similar.There are many reviews on the pseudopotential plane wave method alone or inconnection with the Car–Parrinello algorithm. Older articles 312,157,487,591 as wellas the book by Singh 578 concentrate on the electronic structure part. Other reviews513,472,223,224 present the plane wave method in connection with the moleculardynamics technique.3.1.1 Unit Cell and Plane Wave BasisThe unit cell of a periodically repeated system is defined by the Bravais latticevectors a 1 , a 2 , and a 3 . The Bravais vectors can be combined into a three by threematrix h = [a 1 ,a 2 ,a 3 ] 459 . The volume Ω of the cell is calculated as the determinantof hΩ = deth . (104)Further, scaled coordinates s are introduced that are related to r via hr = hs . (105)43