Ab initio molecular dynamics: Theory and Implementation

The two sets of equations are coupled through the Kohn–Sham energy functionaland special care has to be taken for the integration because of the orthonormalityconstraint.The integrator used in the CPMD code is based on the velocity Verlet / rattlealgorithm 603,638,15 . The velocity Verlet algorithm requires more operations andmore storage than the Verlet algorithm 664 . However, it is much easier to incorporatetemperature control via velocity scaling into the velocity Verlet algorithm. Inaddition, velocity Verlet allows to change the time step trivially and is conceptuallyeasier to handle 638,391 . It is defined by the following equations˙˜R I (t + δt) = Ṙ I (t) + δt2M IF I (t) (231)R I (t + δt) = R I (t) + δt ˙˜R I (t + δt)˙˜c I (t + δt) = ċ I (t) + δt2µ f i(t)˜c i (t + δt) = c i (t) + δt ˙˜c i (t + δt)c i (t + δt) = ˜c i (t + δt) + ∑ jcalculatecalculateF I (t + δt)f i (t + δt)X ij c j (t)Ṙ I (t + δt) = ˙˜R I (t + δt) + δt2M IF I (t + δt)c ˙ ′ i(t + δt) = ˙˜c i (t + δt) + δt2µ f i(t + δt)ċ i (t + δt) = ċ ′ i(t + δt) + ∑ jY ij c j (t + δt) ,where R I (t) and c i (t) are the atomic positions of particle I and the Kohn–Shamorbital i at time t respectively. Here, F I are the forces on atom I, and f i are theforces on Kohn–Sham orbital i. The matrices X and Y are directly related to theLagrange multipliers byX ij = δt22µ Λp ij(232)Y ij = δt2µ Λv ij . (233)Notice that in the rattle algorithm the Lagrange multipliers to enforce the orthonormalityfor the positions Λ p and velocities Λ v are treated as independentvariables. Denoting with C the matrix of wavefunction coefficients c i (G), the or-71