Ab initio molecular dynamics: Theory and Implementation

in terms of a linear combination of basis functions {f ν }, is used in conjunction withan effective one–particle Hamiltonian (such as e.g. in Hartree–Fock or Kohn–Shamtheories). The basis functions might depend explicitly on the nuclear positions (inthe case of basis functions with origin such as atom–centered orbitals), whereas theexpansion coefficients always carry an implicit dependence. This means that fromthe outset two sorts of forces are expected∇ I ψ i = ∑ ν(∇ I c iν ) f ν (r; {R I }) + ∑ νc iν (∇ I f ν (r; {R I })) (66)in addition to the Hellmann–Feynman force Eq. (64).Using such a linear expansion Eq. (65), the force contributions stemming fromthe nuclear gradients of the wavefunction in Eq. (63) can be disentangled into twoterms. The first one is called “incomplete–basis–set correction” (IBS) in solid statetheory 49,591,180 and corresponds to the “wavefunction force” 494 or “Pulay force” inquantum chemistry 494,496 . It contains the nuclear gradients of the basis functionsF IBSI= − ∑ iνµ(〈∇I f ν∣ ∣H NSCe− ɛ i∣ ∣fµ〉+〈fν∣ ∣H NSCe − ɛ i∣ ∣ ∇fµ〉)and the (in practice non–self–consistent) effective one–particle Hamiltonian 49,591 .The second term leads to the so–called “non–self–consistency correction” (NSC) ofthe force 49,591F NSCI∫= −(67)dr (∇ I n) ( V SCF − V NSC) (68)and is governed by the difference between the self–consistent (“exact”) potential orfield V SCF and its non–self–consistent (or approximate) counterpart V NSC associatedto H NSCe ; n(r) is the charge density. In summary, the total force needed in abinitio molecular dynamics simulationsF I = F HFTI+ F IBSI+ F NSCI (69)comprises in general three qualitatively different terms; see the tutorial articleRef. 180 for a further discussion of core vs. valence states and the effect of pseudopotentials.Assuming that self–consistency is exactly satisfied (which is never goingvanishes and H SCFe has to. The Pulay contribution vanishes in the limit of using ato be the case in numerical calculations), the force F NSCIbe used to evaluate F IBSIcomplete basis set (which is also not possible to achieve in actual calculations).The most obvious simplification arises if the wavefunction is expanded in termsof originless basis functions such as plane waves, see Eq. (100). In this case the Pulayforce vanishes exactly, which applies of course to all ab initio molecular dynamicsschemes (i.e. Ehrenfest, Born–Oppenheimer, and Car–Parrinello) using that particularbasis set. This statement is true for calculations where the number of planewaves is fixed. If the number of plane waves changes, such as in (constant pressure)calculations with varying cell volume / shape where the energy cutoff is strictlyfixed instead, Pulay stress contributions crop up 219,245,660,211,202 , see Sect. 4.2. Ifbasis sets with origin are used instead of plane waves Pulay forces arise always andhave to be included explicitely in force calculations, see e.g. Refs. 75,370,371 for suchmethods. Another interesting simplification of the same origin is noted in passing:25