Ab initio molecular dynamics: Theory and Implementation

is used, which represents exactly any reasonable function in the limit of using acomplete set of basis functions. In quantum chemistry, Slater–type basis functions(STOs)f S m(r) = N S m r mxxrymyrz mzexp [−ζ m |r|] (98)with an exponentially decaying radial part and Gaussian–type basis functions(GTOs)f G m(r) = N G m r mxxrymyrz mzexp [ −α m r 2] (99)have received widespread use, see e.g. Ref. 292 for a concise overview–type presentation.Here, N m , ζ m and α m are constants that are typically kept fixed duringa molecular electronic structure calculation so that only the orbital expansion coefficientsc iν need to be optimized. In addition, fixed linear combinations of theabove–given “primitive” basis functions can be used for a given angular momentumchannel m, which defines the “contracted” basis sets.The Slater or Gaussian basis functions are in general centered at the positions ofthe nuclei, i.e. r → r − R I in Eq. (98)–(99), which leads to the linear combinationof atomic orbitals (LCAO) ansatz to solve differential equations algebraically. Furthermore,their derivatives as well as the resulting matrix elements are efficientlyobtained by differentiation and integration in real–space. However, Pulay forces(see Sect. 2.5) will result for such basis functions that are fixed at atoms (or bonds)if the atoms are allowed to move, either in geometry optimization or moleculardynamics schemes. This disadvantage can be circumvented by using freely floatingGaussians that are distributed in space 582 , which form an originless basis set sinceit is localized but not atom–fixed.2.8.2 Plane WavesA vastly different approach has its roots in solid–state theory. Here, the ubiquitousperiodicity of the underlying lattice produces a periodic potential and thus imposesthe same periodicity on the density (implying Bloch’s Theorem, Born–von Karmanperiodic boundary conditions etc., see e.g. Chapt. 8 in Ref. 27 ). This heavilysuggests to use plane waves as the generic basis set in order to expand the periodicpart of the orbitals, see Sect. 3.1.2. Plane waves are defined asf PWG (r) = N exp [iGr] , (100)where the normalization is simply given by N = 1/ √ Ω; Ω is the volume of theperiodic (super–) cell. Since plane waves form a complete and orthonormal set offunctions they can be used to expand orbitals according to Eq. (97), where thelabeling ν is simply given by the vector G in reciprocal space / G–space (includingonly those G–vectors that satisfy the particular periodic boundary conditions). Thetotal electronic energy is found to have a particularly simple form when expressedin plane waves 312 .It is important to observe that plane waves are originless functions, i.e. theydo not depend on the positions of the nuclei {R I }. This implies that the Pulayforces Eq. (67) vanish exactly even within a finite basis (and using a fixed number39