Ab initio molecular dynamics: Theory and Implementation

system dynamics 14 by a sort of dynamical friction mechanism was devised by Noséand Hoover 442,443,444,307 , see e.g. Refs. 12,445,270,585,217 for reviews of this well–established technique. Thereby, the canonical or NV T ensemble is generated inthe case of ergodic dynamics.As discussed in depth in Sect. 2.4, the Car–Parrinello approach to ab initiomolecular dynamics works due to a dynamical separation between the physicaland fictitious temperatures of the nuclear and electronic subsystems, respectively.This separability and thus the associated metastability condition breaks down if theelectronic excitation gap becomes comparable to the thermal energy or smaller, thatis in particular for metallic systems. In order to satisfy nevertheless adiabaticity inthe sense of Car and Parrinello it was proposed to couple separate thermostats 583 tothe classical fields that stem from the electronic degrees of freedom 74,204 . Finally,the (long–term) stability of the molecular dynamics propagation can be increaseddue to the same mechanism, which enables one to increase the time step that stillallows for adiabatic time evolution 638 . Note that these technical reasons to includeadditional thermostats are by construction absent from any Born–Oppenheimermolecular dynamics scheme.It is well–known that the standard Nosé–Hoover thermostat method suffers fromnon–ergodicity problems for certain classes of Hamiltonians, such as the harmonicoscillator 307 . A closely related technique, the so–called Nosé–Hoover–chain thermostat388 , cures that problem and assures ergodic sampling of phase space evenfor the pathological harmonic oscillator. This is achieved by thermostatting theoriginal thermostat by another thermostat, which in turn is thermostatted and soon. In addition to restoring ergodicity even with only a few thermostats in thechain, this technique is found to be much more efficient in imposing the desiredtemperature.Nosé–Hoover–chain thermostatted Car–Parrinello molecular dynamics was introducedin Ref. 638 . The underlying equations of motion readM I ¨R I = −∇ I E KS − M I ˙ξ1 Ṙ I (268)Q n 1 ¨ξ 1 =Q n k¨ξ k =for the nuclear part and[ ]∑M I Ṙ 2 I − gk B T − Q n 1ξ ˙ 1ξ2˙I[Q n ˙ξ]k−1 k−1 2 − k B T − Q n ˙ kξ kξk+1 ˙ (1 − δ kK )µ¨φ i = −HeKS φ i + ∑ ijQ e 1¨η 1 = 2Q e l ¨η l =[ occ ∑iµ 〈φ i |φ i 〉 − T 0 ewhere k = 2, . . ., KΛ ij φ j − µ ˙η 1 ˙φi (269)]− Q e 1 ˙η 1 ˙η 2[Q e l−1 ˙η 2 l−1 − 1 β e]− Q e l ˙η l ˙η l+1 (1 − δ lL )where l = 2, . . ., Lfor the electronic contribution. These equations are written down in density functionallanguage (see Eq. (75) and Eq. (81) for the definitions of E KS and H KSe ,94