Ab initio molecular dynamics: Theory and Implementation

in the previous section. The isobaric–isothermal or NpT ensemble is obtained bycombining barostats and thermostats, see Ref. 389 for a general formulation andRef. 391 for reversible integration schemes.An important practical issue in isobaric ab initio molecular dynamics simulationsis related to problems caused by using a finite basis set, i.e. “incomplete–basis–set” or Pulay–type contributions to the stress, see also Sect. 2.5. Using afinite plane wave basis (together with a finite number of k–points) in the presenceof a fluctuating cell 245,211 one can either fix the number of plane waves or fix theenergy cutoff; see Eq. (122) for their relation according to a rule–of–thumb. Aconstant number of plane waves implies no Pulay stress but a decreasing precisionof the calculation as the volume of the supercell increases, whence leading to a systematicallybiased (but smooth) equation of state. The constant cutoff procedurehas better convergence properties towards the infinite–basis–set limit 245 . However,it produces in general unphysical discontinuities in the total energy and thus in theequation of state at volumes where the number of plane waves changes abruptly,see e.g. Fig. 5 in Ref. 211 .Computationally, the number of plane waves has to be fixed in the frameworkof Car–Parrinello variable–cell molecular dynamics 94,202,201,55 , whereas the energycutoff can easily be kept constant in Born–Oppenheimer approaches to variable–cell molecular dynamics 681,682 . Sticking to the Car–Parrinello technique a practicalremedy 202,55 to this problem consists in modifying the electronic kinetic energyterm Eq. (173) in a plane wave expansion Eq. (172) of the Kohn–Sham functionalE KS Eq. (75)E kin = ∑ ∑ 1f i2 |G|2 |c i (q)| 2 , (283)i qwhere the unscaled G and scaled q = 2πg reciprocal lattice vectors are interrelatedvia the cell h according to Eq. (111) (thus Gr = qs) and the cutoff Eq. (121)is defined as (1/2) |G| 2 ≤ E cut for a fixed number of q–vectors, see Sect. 3.1.The modified kinetic energy at the Γ–point of the Brillouin zone associated to thesupercell reads∑ 1f i ∣ ˜G ( A, σ, Ecut) ∣ eff 2 ∣ |ci (q)| 2 (284)Ẽ kin = ∑ i q∣ ˜G ( A, σ, Ecut) ∣ eff 2 ∣ = |G| 2 + A2{1 + erf[ ]}12 |G|2 − Ecuteffσ(285)where A, σ and Ecut eff are positive definite constants and the number of scaled vectorsq, that is the number of plane waves, is strictly kept fixed.In the limit of a vanishing smoothing (A → 0; σ → ∞) the constant number ofplane wave result is recovered. In limit of a sharp step function (A → ∞; σ → 0)all plane waves with (1/2) |G| 2 ≫ Ecut eff have a negligible weight in Ẽ kin and arethus effectively suppressed. This situation mimics a constant cutoff calculation atan “effective cutoff” of ≈ Ecut eff within a constant number of plane wave scheme. Forthis trick to work note that E cut ≫ Ecut eff has to be satisfied. In the case A > 0 theelectronic stress tensor Π given by Eq. (189) features an additional term (due to99