Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
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Figure 4. (a) Comparison of the x–component of the force acting on one atom of a model systemobtained from Car–Parrinello (solid line) <strong>and</strong> well–converged Born–Oppenheimer (dots) <strong>molecular</strong><strong>dynamics</strong>. (b) Enlarged view of the difference between Car–Parrinello <strong>and</strong> Born–Oppenheimerforces; for further details see text. Adapted from Ref. 467 .the physical variations of the force resolved in Fig. 4(a). These correspond to the“large–amplitude” oscillations of T e visible in Fig. 3 due to the drag of the nucleiexerted on the quasi–adiabatically following electrons having a finite dynamicalmass µ. Note that the inertia of the electrons also dampens artificially the nuclearmotion (typically on a few–percent scale, see Sect. V.C.2 in Ref. 75 for an analysis<strong>and</strong> a renormalization correction of M I ) but decreases as the fictitious massapproaches the adiabatic limit µ → 0. Superimposed on the gross variation in (b)are again high–frequency bound oscillatory small–amplitude fluctuations like for T e .They lead on physically relevant time scales (i.e. those visible in Fig. 4(a)) to “averagedforces” that are very close to the exact ground–state Born–Oppenheimerforces. This feature is an important ingredient in the derivation of adiabatic <strong>dynamics</strong>467,411 .In conclusion, the Car–Parrinello force can be said to deviate at most instants oftime from the exact Born–Oppenheimer force. However, this does not disturb thephysical time evolution due to (i) the smallness <strong>and</strong> boundedness of this difference<strong>and</strong> (ii) the intrinsic averaging effect of small–amplitude high–frequency oscillationswithin a few <strong>molecular</strong> <strong>dynamics</strong> time steps, i.e. on the sub–femtosecond time scalewhich is irrelevant for nuclear <strong>dynamics</strong>.2.4.4 How to Control Adiabaticity ?An important question is under which circumstances the adiabatic separation canbe achieved, <strong>and</strong> how it can be controlled. A simple harmonic analysis of thefrequency spectrum of the orbital classical fields close to the minimum defining theground state yields 467 ( ) 1/2 2(ɛi − ɛ j )ω ij =, (52)µ19