Ab initio molecular dynamics: Theory and Implementation

original unscaled fields φ i (r) do depend on the cell variables h via the normalizationby the cell volume Ω = det h as evidenced by Eq. (275).After these preliminaries a variable–cell extended Lagrangian for ab initio moleculardynamics can be postulated 202,201,55L = ∑ 〈 µ ˙φi (s) ∣ ˙φ〉i (s) − E KS [{φ i }, {hS I }]i+ ∑ ij+ ∑ IΛ ij (〈φ i (s) |φ j (s)〉 − δ ij )12 M I(Ṡt I GṠI)+ 1 2 W Tr ḣt ḣ − p Ω , (278)with additional nine dynamical degrees of freedom that are associated to the latticevectors of the supercell h. This constant–pressure Lagrangian reduces to theconstant–volume Car–Parrinello Lagrangian, see e.g. Eq. (41) or Eq. (58), in thelimit ḣ → 0 of a rigid cell (apart from a constant term p Ω). Here, p defines theexternally applied hydrostatic pressure, W defines the fictitious mass or inertia parameterthat controls the time–scale of the motion of the cell h and the interactionenergy E KS is of the form that is defined in Eq. (75). In particular, this Lagrangianallows for symmetry–breaking fluctuations – which might be necessary to drivea solid–state phase transformation – to take place spontaneously. The resultingequations of motion readM I ¨S I,u = −3∑v=1µ¨φ i (s) = − δEKSδφ ⋆ i (s) + ∑ jWḧuv = Ω3∑s=1∂E KS∂R I,v( ht ) −1vu − M I3∑3∑v=1 s=1G −1uv ˙ G vs Ṡ I,s (279)Λ ij φ j (s) (280)(Πtotus − p δ us) (ht ) −1sv, (281)where the total internal stress tensorΠ totus = 1 ∑ )M I(ṠtΩI GṠI + Π us (282)usIis the sum of the thermal contribution due to nuclear motion at finite temperatureand the electronic stress tensor 440,441 Π which is defined in Eq. (189) and thefollowing equations, see Sect. 3.4.Similar to the thermostat case discussed in the previous section one can recognizea sort of frictional feedback mechanism. The average internal pressure〈(1/3) Tr Π tot 〉 equals the externally applied pressure p as a result of maintainingdynamically a balance between p δ and the instantaneous internal stress Π totby virtue of the friction coefficient ∝ G ˙ in Eq. (279). Ergodic trajectories obtainedfrom solving the associated ab initio equations of motion Eq. (279)–(281) lead toa sampling according to the isobaric–isoenthalpic or NpH ensemble. However, thegenerated dynamics is fictitious similar to the constant–temperature case discussed98