Ab initio molecular dynamics: Theory and Implementation

4.3.2 Many Excited States: Free Energy FunctionalThe free energy functional approach to excited–state molecular dynamics 5,7 is amean–field approach similar in spirit to Ehrenfest molecular dynamics, see Sect. 2.2.The total wavefunction is first factorized into a nuclear and an electronic wavefunctionEq. (3) followed by taking the classical limit for the nuclear subsystem.Thus, classical nuclei move in the average field as obtained from averaging over allelectronic states Eq. (25). A difference is that according to Ehrenfest moleculardynamics the electrons are propagated in real time and can perform non–adiabatictransitions by virtue of direct coupling terms ∝ d kl between all states Ψ k subjectto energy conservation, see Sect. 2.2 and in particular Eqs. (27)–(29). The averageforce or Ehrenfest force is obtained by weighting the different states k according totheir diagonal density matrix elements (that is ∝ |c k (t)| 2 in Eq. (27)) whereas thecoherent transitions are driven by the off–diagonal contributions (which are ∝ c ⋆ k c l,see Eq. (27)).In the free energy approach 5,7 , the excited states are populated according tothe Fermi–Dirac (finite–temperature equilibrium) distribution which is based onthe assumption that the electrons “equilibrate” more rapidly than the timescaleof the nuclear motion. This means that the set of electronic states evolves at agiven temperature “isothermally” (rather than adiabatically) under the inclusionof incoherent electronic transitions at the nuclei move. Thus, instead of computingthe force acting on the nuclei from the electronic ground–state energy it isobtained from the electronic free energy as defined in the canonical ensemble. Byallowing such electronic transitions to occur the free energy approach transcendsthe usual Born–Oppenheimer approximation. However, the approximation of aninstantaneous equilibration of the electronic subsystem implies that the electronicstructure at a given nuclear configuration {R I } is completely independent from previousconfigurations along a molecular dynamics trajectory. Due to this assumptionthe notion “free energy Born–Oppenheimer approximation” was coined in Ref. 101in a similar context. Certain non–equilibrium situations can also be modeled withinthe free energy approach by starting off with an initial orbital occupation patternthat does not correspond to any temperature in its thermodynamic meaning, seee.g. Refs. 570,572,574 for such applications.The free energy functional as defined in Ref. 5 is introduced most elegantly 7,9by starting the discussion for the special case of non–interacting FermionsH s = − 1 2 ∇2 − ∑ IZ I|R I − r|(286)in a fixed external potential due to a collection of nuclei at positions {R I }. Theassociated grand partition function and its thermodynamic potential (“grand freeenergy”) are given byΞ s (µV T) = det 2 (1 + exp [−β (H s − µ)]) (287)Ω s (µV T) = −k B T lnΞ s (µV T) , (288)where µ is the chemical potential acting on the electrons and the square of the101