Ab initio molecular dynamics: Theory and Implementation

3.4 Total Energy, Gradients, and Stress Tensor3.4.1 Total EnergyMolecular dynamics calculations with interaction potentials derived from densityfunctional theory require the evaluation of the total energy and derivatives withrespect to the parameters of the Lagrangian. In this section formulas are given inFourier space for a periodic system. The total energy can be calculated as a sum ofkinetic, external (local and non-local pseudopotential), exchange and correlation,and electrostatic energy (to be compared with Eq. (75))E total = E kin + E PPlocal + EPP nonlocal + E xc + E ES . (172)The individual terms are defined byE kin = ∑ kE PPlocal = ∑ IE PPnonlocal = ∑ kE xc = Ω ∑ G∑ ∑ 1w k2 f i(k) |G + k| 2 |c i (G,k)| 2 (173)∑GE ES = 2π Ω ∑ G≠0iG∆V Ilocal(G) S I (G)n ⋆ (G) (174)∑w k f i (k) ∑i I∑α,β∈I( FαI,i (k) ) ⋆hIαβ F β I,i(k) (175)ɛ xc (G)n ⋆ (G) (176)|n tot (G)| 2G 2 + E ovrl − E self . (177)The overlap between the projectors of the non-local pseudopotential and the Kohn–Sham orbitals has been introduced in the equation aboveFI,i α (k) = √ 1 ∑Pα I (G) S I(G + k) c ⋆ i (G,k) . (178)ΩGAn alternative expression, using the Kohn–Sham eigenvalues ɛ i (k) can also be usedE total = ∑ ∑w k f i (k)ɛ i (k)k i−Ω ∑ (V xc (G) − ε xc (G)) n ⋆ (G)G−2π Ω ∑ |n(G)| 2 − |n c (G)| 2G 2 + E ovrl − E selfG≠0+∆E tot , (179)to be compared to Eq. (86). The additional term ∆E tot in Eq. (179) is needed tohave an expression for the energy that is quadratic in the variations of the chargedensity, as it is true for Eq. (172). Without the correction term, which is zero forthe exact charge density, small differences between the computed and the exact57