Ab initio molecular dynamics: Theory and Implementation

Finally, it is worth commenting in this particular section on a paper entitled“A comparison of Car–Parrinello and Born–Oppenheimer generalized valence bondmolecular dynamics” 229 . In this paper one (computationally expensive) term inthe nuclear equations of motion is neglected 648,405 . It is well known that usinga basis set with origin, such as Gaussians f G ν (r; {R I}) centered at the nuclei, seeEq. (99), produces various Pulay forces, see Sect. 2.5. In particular a linear expansionEq. (65) or (97) based on such orbitals introduces a position dependence intothe orthogonality constraint〈ψ i |ψ j 〉 = ∑ νµ〈 ∣ 〉c ⋆ iνc jµ fG ∣f Gν µ} {{ }= δ ij (73)S νµthat is hidden in the overlap matrix S νµ ({R I }) which involves the basis functions.According to Eq. (44) this term produces a constraint force of the type∑ ∑Λ ij c ⋆ iν c ∂jµ S νµ ({R I }) (74)∂R Iijνµin the correct Car–Parrinello equation of motion for the nuclei similar to the onecontained in the electronic equation of motion Eq. (45). This term has to beincluded in order to yield exact Car–Parrinello trajectories and thus energy conservation,see e.g. Eq. (37) in Ref. 351 for a similar situation. In the case of Born–Oppenheimer molecular dynamics, on the contrary, this term is always absent in thenuclear equation of motion, see Eq. (32). Thus, the particular implementation 229underlying the comparison between Car–Parrinello and Born–Oppenheimer moleculardynamics is an approximate one from the outset concerning the Car–Parrinellopart; it can be argued that this was justified in the early papers 281,282 where thebasic feasibility of both the Hartree Fock– and generalized valence bond–based Car–Parrinello molecular dynamics techniques was demonstrated 285 . Most importantly,this approximation implies that the energy E cons Eq. (48) cannot be rigorously conservedin this particular version of Car–Parrinello molecular dynamics. However,energy conservation of E cons was used in Ref. 229 to compare the efficiency and accuracyof these two approaches to GVB ab initio molecular dynamics (using DIIS forthe Born–Oppenheimer simulations as done in the above–given comparison). Thus,the final conclusion that for “. . . approaches that utilize non–space–fixed bases todescribe the electronic wave function, Born–Oppenheimer AIMD is the method ofchoice, both in terms of accuracy and speed” 229 cannot be drawn from this specificcomparison for the reasons outlined above (independently of the particular basisset or electronic structure method used).The toy system investigated here (see Fig. 5 and Table 1), i.e. 8 silicon atoms ina periodic supercell, is for the purpose of comparing different approaches to ab initiomolecular dynamics quite similar to the system used in Ref. 229 , i.e. clusters of 4 or 6sodium atoms (in addition, qualitatively identical results where reported in Sect. 4for silicon clusters). Thus, it is admissible to compare the energy conservationsreported in Figs. 1 and 2 of Ref. 229 to the ones depicted here in Fig. 5 notingthat the longest simulations reported in Ref. 229 reached only 1 ps. It should be32