A mathematical investigation of the Car-Parrinello method

The Car-Parrinello method for ab-initio molecular dynamics avoids the explicit minimization of energy functionals given by functional density theory in the context of the quantum adiabatic approximation (time-dependent Born-Oppenheimer approximation), Instead, it introduces a fictitious classical dynamics for the electronic orbitals, For many realistic systems this concept allowed first-principle computer simulations for the first time. In this paper we study the quantitative influence of the involved parameter mu, the fictitious electronic mass of the method. In particular, we prove by use of a carefully chosen two-time-scale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order O(mu(1/2)) - provided one starts in the ground state of the electronic system and the electronic excitation spectrum satisfies a certain nondegeneracy condition. Analyzing a two-level model problem we prove that our result cannot be improved in general.