2.1 Quantum Mechanical Treatment of the Many-Particle Problem

The quantum mechanical modeling of a system with N particles of masses m _i leads to the Hamiltonian

Here, V _i( r _i) is an externally given potential in which the ith particle is located and V _ik( r _i; r _k) denotes the interaction potential between the two particles i and k. To analyze or to describe its characteristics, one has to solve the corresponding many-particle Schr dinger equation

where E is the total energy. The wave function depends on the 3N co-ordinates ( configuration space) of all particles:

If we consider nanosystems, most often external potentials are not present and the particles involved are atoms which in turn have to be divided into nuclei (N) and electrons (e). In this case, the interaction potential of Eq. 2.1 is given by the Coulomb potential

where Z is the electron charge number including the sign of the charge.

With a closer look at this many-particle problem, it becomes clear that an exact quantum mechanical solution can probably never be achieved. Here is an example: a relatively small nano-cluster of only 100 argon atoms consists of 100 nuclei and 1800 electrons, which is a total of 1900 particles. In this case, the configuration space consists of 5700 dimensions. The key point for numerical solutions of the Schr dinger equation is the spatial integration. With the assumption that a division of...