Ab initio molecular dynamics: Theory and Implementation

evaluated with some wavefunction Ψ 0 , is certainly a function of the nuclear positions{R I }. But at the same time it can be considered to be a functional of thewavefunction Ψ 0 and thus of a set of one–particle orbitals {ψ i } (or in general ofother functions such as two–particle geminals) used to build up this wavefunction(being for instance a Slater determinant Ψ 0 = det{ψ i } or a combination thereof).Now, in classical mechanics the force on the nuclei is obtained from the derivativeof a Lagrangian with respect to the nuclear positions. This suggests that afunctional derivative with respect to the orbitals, which are interpreted as classicalfields, might yield the force on the orbitals, given a suitable Lagrangian. In addition,possible constraints within the set of orbitals have to be imposed, such as e.g.orthonormality (or generalized orthonormality conditions that include an overlapmatrix).Car and Parrinello postulated the following class of Lagrangians 108L CP = ∑ 12 MIṘ2 I + ∑ 1〈 ∣ 〉 ∣∣2 µ i ˙ψ i ˙ψ i − 〈Ψ 0 |H e |Ψ 0 〉} {{ }Ii} {{ } potential energykinetic energy+ constraints} {{ }(41)orthonormalityto serve this purpose. The corresponding Newtonian equations of motion are obtainedfrom the associated Euler–Lagrange equationsd ∂L= ∂Ldt ∂Ṙ I∂R I(42)d δLdt δ ˙ψ= δLi⋆ δψi⋆ (43)like in classical mechanics, but here for both the nuclear positions and the orbitals;note ψ ⋆ i = 〈ψ i| and that the constraints are holonomic 244 . Following this route ofideas, generic Car–Parrinello equations of motion are found to be of the formM I ¨R I (t) = −∂ 〈Ψ 0 |H e |Ψ 0 〉 +∂ {constraints} (44)∂R I ∂R Iµ i ¨ψi (t) = − δδψ ⋆ i〈Ψ 0 |H e |Ψ 0 〉 + δδψ ⋆ i{constraints} (45)where µ i (= µ) are the “fictitious masses” or inertia parameters assigned to theorbital degrees of freedom; the units of the mass parameter µ are energy times asquared time for reasons of dimensionality. Note that the constraints within thetotal wavefunction lead to “constraint forces” in the equations of motion. Note alsothat these constraintsconstraints = constraints ({ψ i }, {R I }) (46)might be a function of both the set of orbitals {ψ i } and the nuclear positions {R I }.These dependencies have to be taken into account properly in deriving the Car–Parrinello equations following from Eq. (41) using Eqs. (42)–(43), see Sect. 2.5 fora general discussion and see e.g. Ref. 351 for a case with an additional dependenceof the wavefunction constraint on nuclear positions.15