Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
Ab initio molecular dynamics: Theory and Implementation
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The index i runs over all states <strong>and</strong> the states have an occupation f i (k) associatedwith them. The periodic functions u i (r,k) are now exp<strong>and</strong>ed in the plane wavebasisu i (r,k) = √ 1 ∑c i (G,k) exp[iG · r] , (116)ΩG<strong>and</strong> the Kohn–Sham orbitals areφ i (r,k) = √ 1 ∑c i (G,k) exp[i(G + k) · r] , (117)ΩGwhere c i (G,k) are complex numbers. With this expansion the density can also beexp<strong>and</strong>ed into a plane wave basisn(r) = 1 ∑∫dk f i (k) ∑ c ⋆ iΩ(G′ ,k)c i (G,k) exp[i(G + k) · r] (118)G,G ′= ∑ Gin(G) exp[iG · r] , (119)where the sum over G vectors in Eq. (119) exp<strong>and</strong>s over double the range givenby the wavefunction expansion. This is one of the main advantages of the planewave basis. Whereas for atomic orbital basis sets the number of functions neededto describe the density grows quadratically with the size of the system, there isonly a linear dependence for plane waves.3.1.3 K–Points <strong>and</strong> CutoffsIn actual calculations the infinite sums over G vectors <strong>and</strong> cells has to be truncated.Furthermore, we have to approximate the integral over the Brillouin zone by a finitesum over special k–points∫dk → ∑ w k , (120)kwhere w k are the weights of the integration points. Schemes on how to choose theintegration points efficiently are available in the literature 30,123,435 where also anoverview 179 on the use of k–points in the calculation of the electronic structure ofsolids can be found.The truncation of the plane wave basis rests on the fact that the Kohn–Shampotential V KS (G) converges rapidly with increasing modulus of G. For this reason,at each k–point, only G vectors with a kinetic energy lower than a given maximumcutoff12 |k + G|2 ≤ E cut (121)are included in the basis. With this choice of the basis the precision of the calculationwithin the approximations of density functional theory is controlled by oneparameter E cut only.45