Ab initio molecular dynamics: Theory and Implementation

¡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 and 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 and 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 and quantum degrees of freedom is incorporated in bothdirections (at variance with the “classical path” or Mott non–SCF approach todynamics 650,651 ).The approach relying on solving Eq. (15) together with Eq. (18) is sometimescalled “Ehrenfest molecular dynamics” in honor of Ehrenfest who was the first toaddress the question a of how Newtonian classical dynamics 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 molecular dynamicsclearly 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 Standpunkt der Quantenmechanik auf die Newtonschen Grundgleichungender klassischen Mechanik?”8