Ab initio molecular dynamics: Theory and Implementation

where the interaction energy E KS [{φ i } (s) , {R I } (s) ] at time slice s is defined inEq. (75); note that here and in the following the dots denote derivatives with respectto propagation time t and that E0 KS = minE KS . The standard Car–ParrinelloLagrangian, see e.g. Eq. (41) or Eq. (58), is recovered in the limit P = 1 whichcorresponds to classical nuclei. Mixed classical / quantum systems can easily betreated by representing an arbitrary subset of the nuclei in Eq. (319) with only onetime slice.This simplest formulation of ab initio path integrals, however, is insufficient forthe following reason: ergodicity of the trajectories and adiabaticity in the senseof Car–Parrinello simulations are not guaranteed. It is known since the very firstmolecular dynamics computer experiments that quasiharmonic systems (such ascoupled stiff harmonic oscillators subject to weak anharmonicities, i.e. the famousFermi–Pasta–Ulam chains) can easily lead to nonergodic behavior in the samplingof phase space 210 . Similarly “microcanonical” path integral molecular dynamicssimulations might lead to an insufficient exploration of configuration space dependingon the parameters 273 . The severity of this nonergodicity problem is governedby the stiffness of the harmonic intrachain coupling ∝ ω P and the anharmonicity ofthe overall potential surface ∝ E KS /P which establishes the coupling of the modes.For a better and better discretization P the harmonic energy term dominates accordingto ∼ P whereas the mode–mixing coupling decreases like ∼ 1/P. Thisproblem can be cured by attaching Nosé–Hoover chain thermostats 388 , see alsoSect. 4.2, to all path integral degrees of freedom 637,644 .The second issue is related to the separation of the power spectra associatedto nuclear and electronic subsystems during Car–Parrinello ab initio molecular dynamicswhich is instrumental for maintaining adiabaticity, see Sect. 2.4. In abinitio molecular dynamics with classical nuclei the highest phonon or vibrationalfrequency ωnmax is dictated by the physics of the system, see e.g. Fig. 2. This meansin particular that an upper limit is given by stiff intramolecular vibrations whichdo not exceed ωnmax ≤ 5000 cm −1 or 150 THz. In ab initio path integral simulations,on the contrary, ωnmax is given by ω P which actually diverges with increasingdiscretization as ∼ √ P. The simplest counteraction would be to compensate thisartifact by decreasing the fictitious electron mass µ until the power spectra areagain separated for a fixed value of P and thus ω P . This, however, would lead toa prohibitively small time step because ∆t max ∝ √ µ. This dilemma can be solvedby thermostatting the electronic degrees of freedom as well 395,399,644 , see Sect. 4.2for a related discussion in the context of metals.Finally, it is known that diagonalizing the harmonic spring interaction inEq. (319) leads to more efficient propagators 637,644 . One of these transformationand the resulting Nosé–Hoover chain thermostatted equations of motion willbe outlined in the following section, see in particular Eqs. (331)–(337). In additionto keeping the average temperature fixed it is also possible to generate pathtrajectories in the isobaric–isothermal NpT ensemble 646,392 . Instead of usingCar–Parrinello fictitious dynamics in order to evaluate the interaction energy inEq. (318), which is implemented in the CPMD package 142 , it is evident that alsothe Born–Oppenheimer approach from Sect. 2.3 or the free energy functional fromSect. 4.3 can be used. This route eliminates the adiabaticity problem and was taken112