Ab initio molecular dynamics: Theory and Implementation

the Hartree potential V H in Kohn–Sham theory. The exchange operator vanishesexactly if the antisymmetrization requirement of the wavefunction is relaxed, i.e.only the Coulomb contribution survives if a Hartree product is used to representthe wavefunction.The force acting on the orbitals is definedδE HFδψ ⋆ i= H HFe ψ i (96)similarly to Eq. (87). At this stage, the various ab initio molecular dynamicsschemes based on Hartree–Fock theory are defined, see Eqs. (39)–(40) for Born–Oppenheimer molecular dynamics and Eqs. (59)–(60) for Car–Parrinello moleculardynamics. In the case of Ehrenfest molecular dynamics the time–dependentHartree–Fock formalism 162 has to be invoked instead.2.7.4 Post Hartree–Fock TheoriesAlthough post Hartree–Fock methods have a very unfavorable scaling of the computationalcost as the number of electrons increases, a few case studies were performedwith such correlated quantum chemistry techniques. For instance ab initio moleculardynamics was combined with GVB 282,283,228,229,230 , CASSCF 566,567 , as wellas FCI 372 approaches, see also references therein. It is noted in passing that Car–Parrinello molecular dynamics can only be implemented straightforwardly if energyand wavefunction are “consistent”. This is not the case in perturbation theoriessuch as e.g. the widely used Møller–Plesset approach 292 : within standard MP2the energy is correct to second order, whereas the wavefunction is the one given bythe uncorrelated HF reference. As a result, the derivative of the MP2 energy withrespect to the wavefunction Eq. (96) does not yield the correct force on the HFwavefunction in the sense of fictitious dynamics. Such problems are of course absentfrom the Born–Oppenheimer approach to sample configuration space, see e.g.Ref. 328,317,33 for MP2, density functional, and multireference CI ab initio MonteCarlo schemes.It should be kept in mind that the rapidly growing workload of post HF calculations,although extremely powerful in principle, limits the number of explicitelytreated electrons to only a few. The rapid development of correlated electronicstructure methods that scale linearly with the number of electrons will certainlybroaden the range of applicability of this class of techniques in the near future.2.8 Basis Sets2.8.1 Gaussians and Slater FunctionsHaving selected a specific electronic structure method the next choice is relatedto which basis set to use in order to represent the orbitals ψ i in terms of simpleanalytic functions f ν with well–known properties. In general a linear combinationof such basis functionsψ i (r) = ∑ νc iν f ν (r; {R I }) (97)38