Ab initio molecular dynamics: Theory and Implementation

at special values, ξ(R) = ξ ′ , and ˙ξ(R,Ṙ) = 0. The quantity to evaluate is themean forcedFdξ ′ = 〈 Z −1/2 [−λ + k B TG] 〉 ξ ′〈Z−1/2 〉 ξ ′ , (245)where λ is the Lagrange multiplier of the constraint,andG = 1 Z 2 ∑I,JZ = ∑ I1M I M J( ) 21 ∂ξ, (246)M I ∂R I∂ξ∂R I·∂ 2 ξ∂R I ∂R J·∂ξ∂R J, (247)where 〈· · ·〉 ξ ′ is the unconditioned average, as directly obtained from a constrainedmolecular dynamics run with ξ(R) = ξ ′ andF(ξ 2 ) − F(ξ 1 ) =∫ ξ2ξ 1dξ ′dFdξ ′ (248)finally defines the free energy difference. For the special case of a simple distanceconstraint ξ(R) = |R I − R J | the parameter Z is a constant and G = 0.The rattle algorithm, allows for the calculation of the Lagrange multiplier ofarbitrary constraints on geometrical variables within the velocity Verlet integrator.The following algorithm is implemented in the CPMD code. The constraints aredefined byσ (i) ({R I (t)}) = 0 , (249)and the velocity Verlet algorithm can be performed with the following steps.˙˜R I = Ṙ I (t) + δt2M IF I (t)˜R I = R I (t) + δt ˙˜R IR I (t + δt) = ˜R I + δt2 g p (t)2M Icalculate F I (t + δt)Ṙ ′ I = ˙˜R I + δt F I (t + δt)2M IṘ I (t + δt) = Ṙ′ I + δt2M Ig v (t + δt) ,where the constraint forces are defined byg p (t) = − ∑ ig v (t) = − ∑ iλ i ∂σ (i) ({R I (t)})p(250)∂R Iλ i ∂σ (i) ({R I (t)})v. (251)∂R I74