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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¡the Newtonian equation of motion Ṗ I = −∇ I V ({R I }) corresponding to Eq. (11)∫dP I= −∇ I dr Ψ ⋆ H e Ψ ordt∫M I ¨R I (t) = −∇ I dr Ψ ⋆ H e Ψ (15)= −∇ I V Ee ({R I(t)}) (16)can be read off. Thus, the nuclei move according to classical mechanics in aneffective potential Ve E due to the electrons. This potential is a function of only thenuclear positions at time t as a result of averaging H e over the electronic degreesof freedom, i.e. computing its quantum expectation value 〈Ψ|H e |Ψ〉, while keepingthe nuclear positions fixed at their instantaneous values {R I (t)}.However, the nuclear wavefunction still occurs in the TDSCF equation for theelectronic degrees of freedom <strong>and</strong> has to be replaced by the positions of the nuclei forconsistency. In this case the classical reduction can be achieved simply by replacingthe nuclear density |χ({R I }; t)| 2 in Eq. (6) in the limit → 0 by a product of deltafunctions ∏ I δ(R I − R I (t)) centered at the instantaneous positions {R I (t)} of theclassical nuclei as given by Eq. (15). This yields e.g. for the position operator∫dR χ ⋆ →0({R I }; t) R I χ({R I }; t) −→ R I (t) (17)the required expectation value. This classical limit leads to a time–dependent waveequation for the electronsi ∂Ψ∂t = − ∑ 2∇ 2 i2m Ψ + V n−e({r i }, {R I (t)})Ψi e= H e ({r i }, {R I (t)}) Ψ({r i }, {R I }; t) (18)which evolve self–consistently as the classical nuclei are propagated via Eq. (15).Note that now H e <strong>and</strong> thus Ψ depend parametrically on the classical nuclear positions{R I (t)} at time t through V n−e ({r i }, {R I (t)}). This means that feedbackbetween the classical <strong>and</strong> quantum degrees of freedom is incorporated in bothdirections (at variance with the “classical path” or Mott non–SCF approach to<strong>dynamics</strong> 650,651 ).The approach relying on solving Eq. (15) together with Eq. (18) is sometimescalled “Ehrenfest <strong>molecular</strong> <strong>dynamics</strong>” in honor of Ehrenfest who was the first toaddress the question a of how Newtonian classical <strong>dynamics</strong> can be derived fromSchrödinger’s wave equation 174 . In the present case this leads to a hybrid ormixed approach because only the nuclei are forced to behave like classical particles,whereas the electrons are still treated as quantum objects.Although the TDSCF approach underlying Ehrenfest <strong>molecular</strong> <strong>dynamics</strong>clearly is a mean–field theory, transitions between electronic states are includeda The opening statement of Ehrenfest’s famous 1927 paper 174 reads:“Es ist wünschenswert, die folgende Frage möglichst elementar beantworten zu können: WelcherRückblick ergibt sich vom St<strong>and</strong>punkt der Quantenmechanik auf die Newtonschen Grundgleichungender klassischen Mechanik?”8