Ab initio molecular dynamics: Theory and Implementation

In classical molecular dynamics two quantities are conserved during a simulation,the total energy and the total momentum. The same constants of motionapply to (exact) microcanonical Born–Oppenheimer molecular dynamics becausethe only dynamical variables are the nuclear positions and momenta as in classicalmolecular dynamics. In microcanonical Car–Parrinello molecular dynamics thetotal energy of the extended dynamical system composed of nuclear and electronicpositions and momenta, that is E cons as defined in Eq. (48), is also conserved, seee.g. Fig. 3 in Sect. 2.4. There is also a conserved energy quantity in the case of thermostattedmolecular dynamics according to Eq. (268)–(269). Instead of Eq. (48)this constant of motion readsE NVTcons =∑occi++〈 ∣ 〉 ∣∣µ ˙φi ˙φi + ∑ IL∑l=1K∑k=1L12 Qe l ˙η2 l + ∑ η l+ 2Te 0 β η 1e12 Qn k ˙ξ 2 k +l=212 M IṘ2 I + E KS [{φ i }, {R I }]K∑k B Tξ k + gk B Tξ 1 (272)k=2for Nosé–Hoover–chain thermostatted canonical Car–Parrinello molecular dynamics638 .In microcanonical Car–Parrinello molecular dynamics the total nuclear momentumP n is no more a constant of motion as a result of the fictitious dynamics of thewavefunction; this quantity as well as other symmetries and associated invariantsare discussed in Ref. 467 . However, a generalized linear momentum which embracesthe electronic degrees of freedomP CP = P n + P e = ∑ I∑occ〈 ∣ ∣ 〉 ∣∣ ∣∣φiP I + µ ˙φi − ∇r + c.c. (273)ican be defined 467,436 ; P I = M I Ṙ I . This quantity is a constant of motion inunthermostatted Car–Parrinello molecular dynamics due to an exact cancellation ofthe nuclear and electronic contributions 467,436 . As a result, the nuclear momentumP n fluctuates during such a run, but in practice P n is conserved on the average asshown in Fig. 1 of Ref. 436 . This is analogues to the behavior of the physical totalenergy E phys Eq. (49), which fluctuates slightly due to the presence of the fictitiouskinetic energy of the electrons T e Eq. (51).As recently outlined in detail it is clear that the coupling of more than onethermostat to a dynamical system, such as done in Eq. (268)–(269), destroys theconservation of momentum 436 , i.e. P CP is no more an invariant. In unfavorablecases, in particular in small–gap or metallic regimes where there is a substantialcoupling of the nuclear and electronic subsystems, momentum can be transferredto the nuclear subsystem such that P n grows in the course of a simulation. Thisproblem can be cured by controlling the nuclear momentum (using e.g. scaling orconstraint methods) so that the total nuclear momentum P n remains small 436 .96