Ab initio molecular dynamics: Theory and Implementation

The alternative Car–Parrinello formulation seems inconvenient because the singlyand doubly occupied orbitals would have to be constrained not to mix.4.4 Beyond Classical Nuclei4.4.1 IntroductionUp to this point the nuclei were approximated as classical point particles as customarilydone in standard molecular dynamics. There are, however, many situationswhere quantum dispersion broadening and tunneling effects play an important roleand cannot be neglected if the simulation aims at being realistic – which is thegeneric goal of ab initio simulations. The ab initio path integral technique 395 andits extension to quasiclassical time evolution 411 is able to cope with such situationsat finite temperatures. It is also implemented in the CPMD package 142 . The centralidea is to quantize the nuclei using Feynman’s path integrals and at the same timeto include the electronic degrees of freedom akin to ab initio molecular dynamics –that is “on–the–fly”. The main ingredients and approximations underlying the abinitio path integral approach 395,399,644,404 are• the adiabatic separation of electrons and nuclei where the electrons are kept intheir ground state without any coupling to electronically excited states (Born–Oppenheimer or “clamped–nuclei” approximation),• using a particular approximate electronic structure theory in order to calculatethe interactions,• approximating the continuous path integral for the nuclei by a finite discretization(Trotter factorization) and neglecting the indistinguishability of identicalnuclei (Boltzmann statistics), and• using finite supercells with periodic boundary conditions and finite samplingtimes (finite–size and finite–time effects) as usual.Thus, quantum effects such as zero–point motion and tunneling as well as thermalfluctuations are included at some preset temperature without further simplificationsconsisting e.g. in quasiclassical or quasiharmonic approximations, restricting theHilbert space, or in artificially reducing the dimensionality of the problem.4.4.2 Ab Initio Path Integrals: StaticsFor the purpose of introducing ab initio path integrals 395 it is convenient to startdirectly with Feynman’s formulation of quantum–statistical mechanics in termsof path integrals as opposed to Schrödinger’s formulation in terms of wavefunctionswhich was used in Sect. 2.1 in order to derive ab initio molecular dynamics.For a general introduction to path integrals the reader is referred to standard textbooks187,188,334 , whereas their use in numerical simulations is discussed for instancein Refs. 233,126,542,120,124,646,407 .The derivation of the expressions for ab initio path integrals is based on assumingthe non–relativistic standard Hamiltonian, see Eq. (2). The corresponding canonicalpartition function of a collection of interacting nuclei with positions R = {R I }108