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 densit y theory in the context of the quantum adiabatic approximation (time-d ependent Born-Oppenheimer approximation), Instead, it introduces a fic titious classical dynamics for the electronic orbitals, For many reali stic 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 m ethod. In particular, we prove by use of a carefully chosen two-time-s cale asymptotics that the deviation of the Car-Parrinello method from the adiabatic model is of order O(mu(1/2)) - provided one starts in th e 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 gen eral.