Ab initio molecular dynamics: Theory and Implementation

the Car–Parrinello approach 108 , see also Fig. 1. This technique opened novel avenuesto treat large–scale problems via ab initio molecular dynamics and catalyzedthe entire field by making “interesting calculations” possible, see also the closingsection on applications.2.4 Car–Parrinello Molecular Dynamics2.4.1 MotivationA non–obvious approach to cut down the computational expenses of molecular dynamicswhich includes the electrons in a single state was proposed by Car andParrinello in 1985 108 . In retrospect it can be considered to combine the advantagesof both Ehrenfest and Born–Oppenheimer molecular dynamics. In Ehrenfestdynamics the time scale and thus the time step to integrate Eqs. (30) and (31)simultaneously is dictated by the intrinsic dynamics of the electrons. Since electronicmotion is much faster than nuclear motion, the largest possible time stepis that which allows to integrate the electronic equations of motion. Contraryto that, there is no electron dynamics whatsoever involved in solving the Born–Oppenheimer Eqs. (32)–(33), i.e. they can be integrated on the time scale givenby nuclear motion. However, this means that the electronic structure problemhas to be solved self–consistently at each molecular dynamics step, whereas this isavoided in Ehrenfest dynamics due to the possibility to propagate the wavefunctionby applying the Hamiltonian to an initial wavefunction (obtained e.g. by oneself–consistent diagonalization).From an algorithmic point of view the main task achieved in ground–stateEhrenfest dynamics is simply to keep the wavefunction automatically minimizedas the nuclei are propagated. This, however, might be achieved – in principle – byanother sort of deterministic dynamics than first–order Schrödinger dynamics. Insummary, the “Best of all Worlds Method” should (i) integrate the equations ofmotion on the (long) time scale set by the nuclear motion but nevertheless (ii) takeintrinsically advantage of the smooth time–evolution of the dynamically evolvingelectronic subsystem as much as possible. The second point allows to circumventexplicit diagonalization or minimization to solve the electronic structure problemfor the next molecular dynamics step. Car–Parrinello molecular dynamics is an efficientmethod to satisfy requirement (ii) in a numerically stable fashion and makesan acceptable compromise concerning the length of the time step (i).2.4.2 Car–Parrinello Lagrangian and Equations of MotionThe basic idea of the Car–Parrinello approach can be viewed to exploit thequantum–mechanical adiabatic time–scale separation of fast electronic and slownuclear motion by transforming that into classical–mechanical adiabatic energy–scale separation in the framework of dynamical systems theory. In order to achievethis goal the two–component quantum / classical problem is mapped onto a two–component purely classical problem with two separate energy scales at the expenseof loosing the explicit time–dependence of the quantum subsystem dynamics. Furthermore,the central quantity, the energy of the electronic subsystem 〈Ψ 0 |H e |Ψ 0 〉14