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Electronic structure of atoms and molecules

People:  Gero Friesecke
  Heinz-Juergen Flad
  Soeren Behr
  Benjamin Graswald

The chemical behaviour of an atom or molecule relies crucially on its electronic structure. Unlike in many other areas of science, in electronic structure theory a unifying and very precise mathematical model is available, namely the (nonrelativistic, Born-Oppenheimer) time-independent Schrödinger equation, first written down by Schrödinger for the hydrogen atom in 1926, and by Dirac for arbitrary atoms and molecules in 1929. The catch is that the equation is extremely complicated.

One source of complexity is the high-dimensionality of the equation, leading to the so-called problem of exponential scaling: the Schrödinger equation for an atom or molecule with N electrons is a partial differential equation in 3N dimensions, so direct discretization of each coordinate direction into K gridpoints yields K3N gridpoints. Thus the Schrödinger equation for a single Carbon atom (N=6) on a coarse ten point grid in each direction (K=10) already has a prohibitive 1018 degrees of freedom.

Second, understanding electronic structure is a tough multiscale problem: the electronic state of a particular system, and hence its chemical behaviour, is not governed by total energies (mathematically: Schrödinger eigenvalues), but by small energy differences between competing states. Even for single atoms, these differences can already be several orders of magnitude smaller than total energies. For instance the spectral gap between the Carbon atom ground state and the first excited state is only 0.1 percent of the total energy. But this tiny gap is of crucial chemical importance as the two states have different spin and angular momentum symmetry (3P respectively 1D). The angular momentum symmetry of the excited state is that of a transition metal like Scandium or Yttrium, which has entirely different chemical behaviour.

A particular interest of our group is the analysis, design and validation of reduced quantum chemical models which allow to understand and accurately quantify chemical properties of interest with a moderate number of degrees of freedom. One of our innovations is the use, to this end, of rigorous asymptotic analysis of complex models (such as the full Schrödinger equation) in appropriate scaling regimes.