Ab initio molecular dynamics: Theory and Implementation

up e.g. in Refs. 132,37,596,597,428,429,333 .A final observation concerning parallel supercomputers might be useful, see alsoSect. 3.9. It is evident from the Lagrangian Eq. (319) and the resulting equationsof motion (e.g. Eqs. (331)–(337)) that most of the numerical workload comes fromcalculating the ab initio forces on the nuclei. Given a fixed path configurationEq. (316) the P underlying electronic structure problems are independent fromeach other and can be solved without communication on P nodes of a distributedmemory machine. Communication is only necessary to send the final result, essentiallythe forces, to a special node that computes the quantum kinetic contributionto the energy and integrates finally the equations of motions. It is even conceivableto distribute this task on different supercomputers, i.e. “meta–computing” iswithin reach for such calculations. Thus, the algorithm is “embarrassingly parallel”provided that the memory per node is sufficient to solve the complete Kohn–Shamproblem at a given time slice. If this is not the case the electronic structure calculationitself has to be parallelized on another hierarchical level as outlined inSect. 3.9.4.4.3 Ab Initio Path Centroids: DynamicsInitially the molecular dynamics approach to path integral simulations was inventedmerely as a trick in order to sample configuration space similar to the Monte Carlomethod. This perception changed recently with the introduction of the so–called“centroid molecular dynamics” technique 102 , see Refs. 103,104,105,665,505,506,507 forbackground information. In a nutshell it is found that the time evolution of thecenters of mass or centroidsR c I(t) = 1 PP∑s ′ =1R (s′ )I(t) (320)of the closed Feynman paths that represent the quantum nuclei contains quasiclassicalinformation about the true quantum dynamics. The centroid moleculardynamics approach can be shown to be exact for harmonic potentials and to havethe correct classical limit. The path centroids move in an effective potential whichis generated by all the other modes of the paths at the given temperature. Thiseffective potential thus includes the effects of quantum fluctuations on the (quasiclassical)time evolution of the centroid degrees of freedom. Roughly speakingthe trajectory of the path centroids can be regarded as a classical trajectory of thesystem, which is approximately “renormalized” due to quantum effects.The original centroid molecular dynamics technique 102,103,104,105,665 relies onthe use of model potentials as the standard time–independent path integral simulations.This limitation was overcome independently in Refs. 469,411 by combiningab initio path integrals with centroid molecular dynamics. The resulting technique,ab initio centroid molecular dynamics can be considered as a quasiclassical generalizationof standard ab initio molecular dynamics. At the same time, it preservesthe virtues of the ab initio path integral technique 395,399,644,404 to generate exacttime–independent quantum equilibrium averages.113