Introduction to Part I

Abstract

In principle, we can consider any matter, and thus the dynamics of polymers and cells, the interplay of particles. On a quantum mechanical level, Schrödinger’s equation is the basis of description. Here, the particles are electrons and the nuclei of the overall system that interact by electrostatic forces. However, except for very simple systems, there is no analytical solution. Also, since the Schrödinger equation consists in ${{\mathcal{R}}^{{3(N + K)}}}$, where N is the number of electrons and K is the number of nuclei, we encounter the curse of dimensionality. This prohibits any numerical approach for reasons of complexity. Therefore, we have to resort to approximations. A typical approach follows the ideas of Born and Oppenheimer: Since the masses of electrons and nuclei differ by several orders of magnitude, the nuclei evolve classically by Newton’s equations of motion in a potential field which is formed on the quantum mechanical level by the electrons. This results in the molecular dynamics methods of Born and Oppenheimer, Ehrenfest, or Car and Parinello [2], see also [8]. Here, in every time step, an electronic Schrödinger equation has to be solved approximatively, e.g., by the density functional method or the Hartree-Fock approach. Since this is still quite costly and can only be done for systems of moderate size, a further approximation is used: The effect of the electrons is expressed by analytical potential functions whose forms depend on a set of parameters. These parameters are fitted either to measurements or to the results of ab initio calculations for model situations. Thus, empirical force fields are obtained in which the nuclei now evolve over time according to Newton’s equations. This is the basis of the classical molecular dynamics method.