Ab initio molecular dynamics: Theory and Implementation

ing numericallyM I ¨R I (t) = −∇ I∫dr Ψ ⋆ H e Ψ= −∇ I 〈Ψ |H e |Ψ〉 (25)= −∇ I 〈H e 〉= −∇ I VeE[i ∂Ψ∂t = − ∑ ]2∇ 2 i2m + V n−e({r i }, {R I (t)}) Ψi e= H e Ψ (26)the coupled set of equations simultaneously. Thereby, the a priori constructionof any type of potential energy surface is avoided from the outset by solving thetime–dependent electronic Schrödinger equation “on–the–fly”. This allows one tocompute the force from ∇ I 〈H e 〉 for each configuration {R I (t)} generated by moleculardynamics; see Sect. 2.5 for the issue of using the so–called “Hellmann–Feynmanforces” instead.The corresponding equations of motion in terms of the adiabatic basis Eq. (20)and the time–dependent expansion coefficients Eq. (19) read 650,651M I ¨R I (t) = − ∑ |c k (t)| 2 ∇ I E k − ∑kk,l∑i ċ k (t) = c k (t)E k − iI,lc ⋆ k c l (E k − E l )d klI (27)c l (t)Ṙ I d klI , (28)where the coupling terms are given byd klI ({R I (t)}) =∫dr Ψ ⋆ k∇ I Ψ l (29)with the property d kkI≡ 0. The Ehrenfest approach is thus seen to include rigorouslynon–adiabatic transitions between different electronic states Ψ k and Ψ l withinthe framework of classical nuclear motion and the mean–field (TDSCF) approximationto the electronic structure, see e.g. Refs. 650,651 for reviews and for instanceRef. 532 for an implementation in terms of time–dependent density functional theory.The restriction to one electronic state in the expansion Eq. (19), which is inmost cases the ground state Ψ 0 , leads toM I ¨R I (t) = −∇ I 〈Ψ 0 |H e | Ψ 0 〉 (30)i ∂Ψ 0∂t= H e Ψ 0 (31)as a special case of Eqs. (25)–(26); note that H e is time–dependent via the nuclearcoordinates {R I (t)}. A point worth mentioning here is that the propagation of thewavefunction is unitary, i.e. the wavefunction preserves its norm and the set oforbitals used to build up the wavefunction will stay orthonormal, see Sect. 2.6.11