Ab initio molecular dynamics: Theory and Implementation

Figure 3. Various energies Eqs. (48)–(51) for a model system propagatedvia Car–Parrinellomoleculardynamics for at short (up to 300 fs), intermediate, and long times (up to 6.3 ps); for furtherdetails see text. Adapted from Ref. 467 .for the stability of the Car–Parrinello dynamics, vide infra. But already the visiblevariations are three orders of magnitude smaller than the physically meaningful oscillationsof V e . As a result, E phys defined as E cons − T e or equivalently as the sumof the nuclear kinetic energy and the electronic total energy (which serves as thepotential energy for the nuclei) is essentially constant on the relevant energy andtime scales. Thus, it behaves approximately like the strictly conserved total energyin classical molecular dynamics (with only nuclei as dynamical degrees of freedom)or in Born–Oppenheimer molecular dynamics (with fully optimized electronic degreesof freedom) and is therefore often denoted as the “physical total energy”.This implies that the resulting physically significant dynamics of the nuclei yieldsan excellent approximation to microcanonical dynamics (and assuming ergodicityto the microcanonical ensemble). Note that a different explanation was advocatedin Ref. 470 (see also Ref. 472 , in particular Sect. VIII.B and C), which was howeverrevised in Ref. 110 . A discussion similar in spirit to the one outlined here 467 isprovided in Ref. 513 , see in particular Sect. 3.2 and 3.3.Given the adiabatic separation and the stability of the propagation, the centralquestion remains if the forces acting on the nuclei are actually the “correct” onesin Car–Parrinello molecular dynamics. As a reference serve the forces obtainedfrom full self–consistent minimizations of the electronic energy min {ψi}〈Ψ 0 |H e |Ψ 0 〉at each time step, i.e. Born–Oppenheimer molecular dynamics with extremely wellconverged wavefunctions. This is indeed the case as demonstrated in Fig. 4(a):the physically meaningful dynamics of the x–component of the force acting on onesilicon atom in the model system obtained from stable Car–Parrinello fictitiousdynamics propagation of the electrons and from iterative minimizations of the electronicenergy are extremely close.Better resolution of one oscillation period in (b) reveals that the gross deviationsare also oscillatory but that they are four orders of magnitudes smaller than18